Precalculus with Geometry and Trigonometry

Precalculus with Geometry and Trigonometry by Avinash Sathaye, Professor of Mathematics 1 Department of Mathematics, University of Kentucky Āryabhaṭa This book may be freely downloaded for personal use from the author s web site www.msc.uky.edu/sohum/ma110/text/ma110 fa08.pdf Any commercial use must be preauthorized by the author. Send an email to sathaye@uky.edu for inquiries. November 23, 2009 1 Partially supported by NSF grant thru AMSP(Appalachian Math Science Partnership)

ii Introduction. This book on Precalculus with Geometry and Trigonometry should be treated as simply an enhanced version of our book on College Algebra. Most of the topics that appear here have already been discussed in the Algebra book and often the text here is a verbatim copy of the text in the other book. We expect the student to already have a strong Algebraic background and thus the algebraic techniques presented here are more a refresher course than a first introduction. We also expect the student to be heading for higher level mathematics courses and try to supply the necessary connections and motivations for future use. Here is what is new in this book. In contrast with the Algebra book, we make a more extensive use of complex numbers. We use Euler s representation of complex numbers as well as the Argand diagrams extensively. Even though these are described and shown to be useful, we do not yet have tools to prove these techniques properly. They should be used as motivation and as an easy method to remember the trigonometric results. We have supplied a brief introduction to matrices and determinants. The idea is to supply motivation for further study and a feeling for the Linear Algebra. In the appendix, we give a more formal introduction to the structure of real numbers. While this is not necessary for calculations in this course, it is vital for understanding the finer concepts of Calculus which will be introduced in higher courses. We have also included an appendix discussing summation of series - both finite and infinite, as well as a discussion of power series. While details of convergence are left out, this should generate familiarity with future techniques and a better feeling for the otherwise mysterious trigonometric and exponential functions.

Contents 1 Review of Basic Tools. 1 1.1 Underlying field of numbers......................... 1 1.1.1 Working with Complex Numbers................. 7 1.2 Indeterminates, variables, parameters.................. 12 1.3 Basics of Polynomials........................... 13 1.3.1 Rational functions......................... 17 1.4 Working with polynomials........................ 17 1.5 Examples of polynomial operations.................... 21 Example 1. Polynomial operations................... 21 Example 2. Collecting coefficients.................... 22 Example 3. Using algebra for arithmetic................ 22 Example 4. The Binomial Theorem................... 23 Example 5. Substituting in a polynomial................ 27 Example 6. Completing the square................... 28 2 Solving linear equations. 31 2.1 What is a solution?............................ 32 2.2 One linear equation in one variable.................... 34 2.3 Several linear equations in one variable.................. 35 2.4 Two or more equations in two variables................. 36 2.5 Several equations in several variables................... 37 2.6 Solving linear equations efficiently.................... 38 Example 1. Manipulation of equations................. 38 Example 2. Cramer s Rule........................ 39 Example 3. Exceptions to Cramer s Rule................ 41 Example 4. Cramer s Rule with many variables............ 42 3 The division algorithm and applications 45 3.1 Division algorithm in integers....................... 45 Example 1. GCD calculation in Integers................ 48 3.2 Āryabhaṭa algorithm: Efficient Euclidean algorithm........... 49 Example 2. Kuttaka or Chinese Remainder Problem......... 52 1

2 CONTENTS Example 3. More Kuttaka problems.................. 53 Example 4. Disguised problems..................... 54 3.3 Division algorithm in polynomials.................... 56 3.4 Repeated Division............................. 60 3.5 The GCD and LCM of two polynomials................. 62 Example 5. Āryabhaṭa algorithm for polynomials........... 64 Example 6. Efficient division by a linear polynomial.......... 67 Example 7. Division by a quadratic polynomial............ 69 4 Introduction to analytic geometry. 73 4.1 Coordinate systems............................ 73 4.2 Geometry: Distance formulas....................... 75 4.2.1 Connection with complex numbers................ 77 4.3 Change of coordinates on a line...................... 78 4.4 Change of coordinates in the plane.................... 79 4.5 General change of coordinates....................... 81 4.5.1 Description of Isometies...................... 81 4.5.2 Complex numbers and plane transformations.......... 82 4.5.3 Examples of complex transformations.............. 82 4.5.4 Examples of changes of coordinates:............... 83 5 Equations of lines in the plane. 87 5.1 Parametric equations of lines....................... 87 Examples. Parametric equations of lines................ 89 5.2 Meaning of the parameter t:....................... 91 Examples. Special points on parametric lines.............. 93 5.3 Comparison with the usual equation of a line.............. 95 5.4 Examples of equations of lines....................... 100 Example 1. Points equidistant from two given points......... 102 Example 2. Right angle triangles.................... 103 6 Special study of Linear and Quadratic Polynomials. 105 6.1 Linear Polynomials............................. 105 6.2 Factored Quadratic Polynomial...................... 107 Interval notation. Intervals on real line................ 107 6.3 The General Quadratic Polynomial.................... 110 6.4 Examples of Quadratic polynomials................... 113 7 Functions 115 7.1 Plane algebraic curves.......................... 115 7.2 What is a function?............................ 117 7.3 Modeling a function............................ 120 7.3.1 Inverse Functions.......................... 124

CONTENTS 3 8 The Circle 129 8.1 Circle Basics................................ 129 8.2 Parametric form of a circle........................ 131 8.3 Application to Pythagorean Triples.................... 133 Pythagorean Triples. Generation of.................. 133 8.4 Examples of equations of a circle..................... 137 Example 1. Intersection of two circles.................. 138 Example 1a. Complex intersection of two circles............ 140 Example 2. Line joining through the intersection of two circles.... 141 Example 3. Circle through three given points............. 142 Example 4. Exceptions to a circle through three points........ 142 Example 5. Smallest circle with a given center meeting a given line. 143 Example 6. Circle with a given center and tangent to a given line.. 144 Example 7. The distance between a point and a line......... 145 Example 8. Half plane defined by a line................ 147 9 Trigonometry 149 9.1 Trigonometric parameterization of a circle................ 149 Definition. Trigonometric Functions................... 153 9.2 Basic Formulas for the Trigonometric Functions............. 157 9.3 Connection with the usual Trigonometric Functions........... 161 9.4 Important formulas............................. 162 9.5 Using trigonometry............................. 174 9.6 Proofs I................................... 180 9.7 Proofs II.................................. 181 10 Looking closely at a function 187 10.1 Introductory examples........................... 187 Parabola. Analysis near its points.................... 187 Circle. Analysis near its points...................... 191 10.2 Analyzing a general curve y = f(x) near a point (a, f(a))....... 192 10.3 The slope of the tangent, calculation of the derivative......... 194 10.4 Derivatives of more complicated functions................ 198 10.5 General power and chain rules....................... 199 10.6 Using the derivatives for approximation................. 202 Linear Approximation. Examples................... 203 11 Root finding 209 11.1 Newton s Method............................. 209 11.2 Limitations of the Newton s Method.................. 211

4 CONTENTS 12 Appendix 213 12.1 An analysis of 2 as a real number.................... 213 12.2 Idea of a Real Number........................... 215 12.3 Summation of series............................ 217 12.3.1 A basic formula.......................... 218 12.3.2 Using the basic formula...................... 218 12.4 On the exponential and logarithmic functions.............. 222 12.5 Infinite series and their use........................ 225 12.6 Inverse functions by series........................ 233 12.7 Decimal expansion of a Rational Number................ 238 12.8 Matrices and determinants: a quick introduction............ 241

Chapter 1 Review of Basic Tools. 1.1 Underlying field of numbers. Mathematics may be described as the science of manipulating numbers. The process of using mathematics to analyze the physical universe often consists of representing events by a set of numbers, converting the laws of physical change into mathematical functions and equations and predicting or verifying the physical events by evaluating the functions or solving the equations. We begin by describing various types of numbers in use. During thousands of years, mathematics has developed many systems of numbers. Even when some of these appear to be counterintuitive or artificial, they have proved to be increasingly useful in developing advanced solution techniques. To be useful, our numbers must have a few fundamental properties. We should be able to perform the four basic operations of algebra: addition subtraction multiplication and division (except by 0) and produce well defined numbers as answers. Any set of numbers having all these properties is said to be a field of numbers (or constants). Depending on our intended use, we work with different fields of numbers. Here is a description of fields of numbers that we typically use. 1. Rational numbers Q. The most natural idea of numbers comes from simple counting 1, 2, 3, and these form the set of natural numbers often denoted by IN. 1

2 CHAPTER 1. REVIEW OF BASIC TOOLS. These are not yet good enough to make a field since subtraction like 2 5 is undefined. To fix the subtraction property, we can add in the zero 0 and negative numbers. 1 This produces the set of integers, Z = {0, ±1, ±2, }. These are still deficient because the division does not work. You cannot divide 1 by 2 and get an integer back. The natural next step is to introduce the fractions m where m, n are integers n and n 0. You probably remember the explanation in terms of picking up parts; thus 3 -th of a pizza is three of the eight slices of one pizza. 8 We now have a natural field at hand, the so-called field of rational numbers Q = { m m, n Z, n 0} n 2. Real numbers R. One familiar way of thinking of numbers is as decimal numbers, say something like 2.34567 which is nothing but a rational number 2 whose denominator is a power of 10. 234567 100,000 1 This simple sounding idea took several hundred years to develop and be accepted, because the idea of a negative count is hard to imagine. If we think of a number representing money owned, then a negative number can easily be thought of as money owed! The concept of negative numbers and zero was developed and expanded in India where a negative number is called ṛṇa which also means debt! The word used for a positive number, is similarly dhana which means wealth. The point is that even though the idea of certain numbers sounds unrealistic or impossible, one should keep an open mind and accept and use them as needed. They can be useful and somebody may find a good interpretation for them some day. 2 Indeed, the number 10 can and is often replaced by other convenient numbers. The computer scientists often prefer 2 in place of 10, leading to the binary numbers, or they also use 8 or 16 in other contexts, leading to octal or hexadecimal numbers. In Number Theory, it is customary to replace 10 by some prime p and study the resulting p-adic numbers. It is also possible and sometimes convenient to choose a product of suitable numbers, rather than power of a single one. It is interesting to consider numbers of the form a 1 + a 2 2! + a 3 3! + where a 1 is an arbitrary integer, while 0 a 2 < 2, 0 a 3 < 3 and so on. Thus, you can verify that: 17 9 = 1 + 1 2! + 2 3! + 1 4! + 1 5! + 4 6!

1.1. UNDERLYING FIELD OF NUMBERS. 3 Thus, rational numbers whose denominators are powers of 10 are called decimal numbers. A calculator, especially a primitive one, deals exclusively with such numbers. One quickly realizes that even something as simple as 1 runs into problems if 3 we insist on using only decimal numbers. It has successive approximations 0.3, 0.33, 0.333, but no finite decimal will ever give the exact value 1. 3 While this is somewhat disturbing, we do have the choice of writing 0.3 with the understanding that it is the decimal number obtained by repeating the digit 3 under the bar indefinitely. It can be shown that any rational number m can be written as a repeating n decimal which consists of an infinite decimal number with a certain group of digits repeating indefinitely from some point on. For example, 1 = 0.142857142857 = 0.142857. 7 For a detailed explanation of why all rationals can be written as repeating decimals, please see the appendix. Thus, we realize that if we are willing to handle infinite decimals, we have all the rational numbers and as a bonus we get a whole lot of new numbers whose decimal expansions don t repeat. What do we get? We get the so-called field of real numbers. Because we are so familiar with the decimals in everyday life, these feel natural and easy, except for the fact that almost always we are dealing with approximations and issues of limiting values. Thus for example, there cannot be any difference between the decimal 0.1 and the decimal 0.09999 = 0.09 even though they appear different! The ideas that the sequence of longer and longer decimals 0.09, 0.099, 0.0999, gets closer and closer to the decimal 0.1 and we can precisely formulate a meaning for the infinite decimal as a limit equal to 0.1 are natural, but rather sophisticated. While these were informally being used all the time, they were formalized only in the last four hundred years. The subject of Calculus pushes them to their natural logical limit. You might recall that the set of real numbers can be represented as points of the number line by the following procedure: and it can be thought of as represented by 1 12114. We may call this the factorial system. This idea has the advantage of keeping all rationals as finite expressions, but is clearly not as easy to use as the decimal system!

4 CHAPTER 1. REVIEW OF BASIC TOOLS. Mark some convenient point as the origin associated with zero. Mark some other convenient point as the unit point, identified with 1. We define its distance from the origin to be the unit distance or simply the unit. Then every decimal number gets an appropriate position on the line with its distance from the origin set equal to its absolute value and is placed on the correct side of the zero depending on its sign. Thus the number 2 gets marked in the same direction as the unit point but at twice the unit distance. The number 3 gets marked on the opposite side of the unit point and at three times the unit distance. In general, a positive number x is on the same side of the unit point at distance equal x times the unit distance and the point x is on the other side at the same distance. See some illustrated points below. 3 1 0 1 2 3.125 4.25 It seems apparent that the real numbers must fill up the whole number line and we should not need any further numbers. Actually, this feeling is misleading, since you can get the same feeling by plotting lots of rational points on the number line and see it visibly fill up! It takes an algebraic manipulation and some clever argument to show that the positive square root of 2, usually denoted as 2 cannot be a rational number and yet deserves its rightful position on the number line. If you have not thought about it, it is a very good challenge to prove that there are no positive integers m, n which will satisfy: 3 2 = m n in other words m2 = 2n 2. It can, however, be proved that in some sense the real numbers (or the set of all decimal numbers, allowing infinite decimals) do fill up the number line. The precise proof of this fact will be presented in a good first course in Calculus. It would seem natural to be satisfied that we have found all the numbers that need to be found and can stop at R. However, we now show that there is something more interesting to find! 3 In the appendix, you will find a precise argument to show how 2 can be given a proper place on the real number line.

1.1. UNDERLYING FIELD OF NUMBERS. 5 3. Complex numbers. C. Rather than the geometric idea of filling up the number line, we could think of numbers as needed solutions of natural equations. Thus, the negative integer 2 is needed to be able to solve x + 5 = 3. The fraction 1 is needed to be able to solve 2x = 1. 2 The number 2 is needed to solve x 2 = 2. In fact, in a beginning Calculus course you will meet a proof that every polynomial equation: of odd degree n has a real solution! x n + a 1 x n 1 + a n = 0 Thus, real numbers provide solutions to many polynomial equations, but miss one important equation: x 2 + 1 = 0. Why does it not have a real solution? Suppose we name a solution of the equation as i and try to think where to put it on the real number line. Note that i 2 = 1. If i were positive, then i i = i 2 would be positive, but it is 1 and hence negative. Similarly, if i were negative, then again i 2 would be positive since a product of two negative numbers is positive. But it is 1 and hence negative. Thus, either assumption leads to a contradiction! Thus, i cannot be put on the left or the right side of 0. This number i is certainly not zero, for otherwise its square would be zero! Thus, i has to be somewhere outside the real number line and we have found a new number that we must have, if we want to solve all polynomial equations. As a result, we have a whole new set of numbers of the form a + bi where a, b are any real numbers. Do these already form a field? Surprisingly, they do! You only need to verify the following formulas by comparing both sides of the equations, by cross multiplication, if necessary. (a + ib)(c + id) = (ac bd) + i(ad + bc) and 1 (a + ib) = a (a 2 + b 2 ) b (a 2 + b 2 ) i. Thus we have a new field at hand called the field of complex numbers and denoted by C.

6 CHAPTER 1. REVIEW OF BASIC TOOLS. Its points can be conveniently represented as points of a plane by simply plotting the complex number a + bi at the point P(a, b) the point with real coordinates a, b. Here is a complex number plane with some points plotted. Such pictures are called Argand diagrams. The complex number plane is also called just the complex line by some people. i 1+i 0 1/(1+i) 2-i To verify your understanding of complex numbers, assign yourself the task of verifying the following: 4 1 1 + i = 1 i 2, ( 1 ± 3i) 3 = 8, (1 + i) 4 = 4. 4 We illustrate how you would check the first of these. By cross multiplying, we see that we want to show: 2(1) = (1 + i)(1 i). By simple expansion the right hand side (usually shortened to RHS) is (1)(1 i) + (i)(1 i) = 1 i + i i 2 = 1 i 2. Using the basic fact i 2 = 1, we see that this becomes 1 + 1 = 2 or the left hand side (LHS). Done!

1.1. UNDERLYING FIELD OF NUMBERS. 7 One of the most satisfying properties of complex numbers is known as the Fundamental Theorem of Algebra It states that every polynomial equation with complex coefficients: x n + a 1 x n 1 + + a n = 0 has a complex solution! Of course, the real numbers are included in complex numbers, so the coefficients can be all real! Many clever proofs of this theorem are known, but the problem of actually finding the solution to a given polynomial equation is hard and often has to be left as an approximate procedure. Only equations of degree 1 and 2 have well known formulas to find roots, equations of degrees 3 and 4 have known formulas, but they are rather messy and for higher degrees, it has been established that there cannot be any simple minded formulas. Only suitable procedures for finding approximate solutions to a desired accuracy can be established. We may, occasionally, denote our set of underlying numbers by the letter F (to remind us of its technical mathematical term - a field). Usually, we may just call them numbers and we will usually be talking about the field of real numbers i.e. F = R, but other fields are certainly possible. There are some fields which have only finitely many numbers, but have rich algebraic properties. These are known as finite fields and have many applications. From time to time, we may invite you to consider these number fields and imagine what will happen to our results when you switch over to them. However, they will not form an integral part of this course. 1.1.1 Working with Complex Numbers. We list a few useful definitions about complex numbers which help in calculations. 1. Real and Imaginary Parts. Let z = x + iy be a complex number so that x, y are real. Then x is called its real part and is denoted as Re(z). Similarly, y is called its imaginary part and is denoted as Im(z). It is important to note that the imaginary part itself is very real and the i is not included in it. Thus, any complex number z is always equal to Re(z) + iim(z). 2. Complex conjugate. Let z = x + iy be a complex number so that x, y are real. Then we define its complex conjugate (or just called the conjugate) by z = x iy = Re(z) iim(z).

8 CHAPTER 1. REVIEW OF BASIC TOOLS. This works without simplifying z. Thus, if z = 1 + i 1 i then z = 1 i 1 + i. Thus, we do not have to explicitly write it as x + iy to find its conjugate. Computing conjugates. It is easy to check that given complex numbers z, w we have: ( z z + w = z + w, zw = z w and if w 0 then = w) z w. In turn, the conjugate can help us figure out the real and imaginary parts thus: Thus, for z = 1+i 1 i Re(z) = z + z 2 as above, we get: Re(z) = 1 2 and Im(z) = z z. 2i ( 1+i + ) ( 1 i 1 i 1+i ) = 1 (1 2i+i 2 )+(1+2i+i 2 ) 2 1 i 2 = 1 2 (2+2i2 1 i 2 ) = 0. Also, Im(z) = ( 1 1+i ) 1 i 2i ( 1 i 1+i ) = 1 (1 2i+i 2 ) (1+2i+i 2 ) 2i 1 i 2 = 1 2i ( 4i 1 i 2 ) = 2. Thus, our z was simply Re(z) + iim(z) = 0 2i = 2i. 3. Absolute Value. Recall that a real number x has an absolute value x which is defined as: x = x if x 0 and x otherwise. It is important to keep in mind that x represents the distance to the origin from x. If x 0, then x = ±1. We may simply declare this to be the direction x (from 0) of a non zero number x. Note that this direction evaluates to 1 if the number x is positive and 1 if it is negative. Naturally, we have an absolute value for a complex number z = x + iy defined as: z = x 2 + y 2 and this is clearly a non negative real number.

1.1. UNDERLYING FIELD OF NUMBERS. 9 What takes place of the sign? Since the complex numbers live in the plane, there are infinitely many directions available to them. Any complex number z with z = 1 represents a direction from the origin to it and all such numbers lie on the unit circle defined by x 2 + y 2 = 1. 5 As before, we can compute the absolute value more efficiently by noting: zz = x 2 i 2 y 2 = x 2 + y 2 = z 2 or, simply z = zz. We note that z is in fact a real number and this is simply the usual square root of the non negative number zz! We may find it useful to declare z as the length of the complex number z. As before, when z 0 is a complex number, then we declare z to be its z direction (from 0). 4. Euler Representation of a Complex Number. There is a very convenient way of thinking of complex numbers and calculating with them which often attributed to Euler. We don t have enough tools to give its complete proof, but it is very useful to learn to use it and learn the proof later. As noted above, every complex number z = x + iy such that z = 1 lies on the unit circle x 2 + y 2 = 1. It would be convenient to describe all these points by convenient single real numbers. We simply take the angle for z to be the measure of the arclength s measured along the unit circle from the point corresponding to the number 1 to the point corresponding to the number z. See the picture below. 5 This is consistent with the real definition. Since a real number can be thought of as a complex number with y = 0 the only real numbers with unit length are x with x 2 = 1. Thus we get x = ±1, giving the two directions from origin!

10 CHAPTER 1. REVIEW OF BASIC TOOLS. z B Angle for z is the arc AB O A 1 Note that the arclength of the whole circle is 2π and thus adding any multiple of 2π to s will land us in the same point as z. For a complex number z with z = 1, we define its argument to be the arclength s from 1 to z measured counter clockwise. Formally, we define Arg(z) = s, the principal value of the argument which is a non negative real number less than 2π. The general values of the argument are said to be any members of the set {s + 2nπ} where n is an integer. This set is called arg(z). Finally, when z is any non zero complex number, then we define Arg(z) = Arg(w) where w is complex number z of length 1 in the direction z of z. The Euler representation of a complex number states that any non zero complex number z can be uniquely written as: This requires some explanation. z = r exp(iθ) where r = z and θ = Arg(z). The exponential function exp(t) can be formally defined as exp(t) = 1 + t 1! + t2 2! + + tn n! +. The right hand side makes complete sense if we add up terms up to some large enough value n. It takes much deeper analysis to define the infinite sum. Here, we shall only worry about the formal algebraic properties, leaving the finer details for future study.

1.1. UNDERLYING FIELD OF NUMBERS. 11 5. We will now show how the series for exp(t) is related to the following series for the important sine and cosine functions: cos(t) = 1 t2 2! + t4 t2n + + ( 1)n 4! (2n)! + and sin(t) = t 1! t3 3! + + ( 1)n 1 t 2n 1 (2n 1)! +. 6. Note that: exp(it) = ( 1 + it = 1 + i2 t 2 = + (it)2 + (it)3 + 1! 2! 3! ) + i4 t 4 + + i2n t 2n + 2! 4! (2n)! ( ) it + i3 t 3 + + i2n 1 t 2n 1 + 1! 3! (2n 1)! + ( 1 t2 ( +i + t4 2! 4! t 1! t3 3! To understand the above work, note that + + ( 1)n t2n (2n)! + ) + + ( 1)n 1 t2n 1 (2n 1)! + ). i 2n = (i 2 ) n = ( 1) n and i 2n 1 = (i)(i 2n 2 ) = i(i 2 ) n 1 = i( 1) n 1. In higher mathematics, the trigonometric functions sin(t), cos(t) (that you may have seen in high school) are defined by: This means: 6 cos(t) = Re(exp(it)) and sin(t) = Im(exp(it)). exp(it) =cos(t) + i sin(t) 7. It is instructive to use your calculator to compare the two sides of this formula. Evaluate the value of say sin(1) and cos(1) directly by using the calculator key. (Be sure to put the calculator in the radian mode.) Then calculate the 6 This leads to the well known formula e iπ = 1 which connects three famous numbers e = exp(1), i and π with the usual 1.

12 CHAPTER 1. REVIEW OF BASIC TOOLS. sum of several terms of this formula and compare how the answers change. Of course, when a calculator gives an answer, it is subject to approximation error in evaluation as well as addition, so don t expect precise agreement! An infinite precision computer program (like Maple) might be more suited for this exercise. The main point of this discussion is this: Euler s formulas give a completely precise definition of complicated functions sin(t), cos(t) with built in polynomial approximations. The final aim of Calculus is to push this technique to more and more functions, ideally to all well behaved functions. 1.2 Indeterminates, variables, parameters It is useful to set up some precise meaning of usual algebraic terms to avoid future misunderstanding. If you see an expression like ax 2 + bx + c; you usually think of this as an expression in a variable x with constants a, b, c. At this stage, x as well as a, b, c are unspecified. However, we are assuming that a, b, c are fixed while x is a variable which may be pinned down after some extra information - for example, if the expression is set equal to zero. Thus, in algebraic expressions, the idea of who is constant and who is a variable is a matter of declaration or convention. Variables A variable is a symbol, typically a Greek or Standard English letter, used to represent an unknown quantity. However, not all symbols are variables and typically a simple declaration tells us which is a variable and which is a constant. Take this familiar expression for example: mx + c. On the face of it, all the letters m, c, x are potential variables. You may already remember seeing this in connection with the equation of a line. Recall that m in mx + c used to be a constant and not a variable. The letter m was the slope of a certain line and was a fixed number when we knew the line. The letter c in mx + c also used to be a constant. It used to be the y-intercept (i.e. the y-coordinate of the intersection of the line and the y-axis). The x in mx + c is the variable in this expression and represents a changing value of the x-coordinate. The expression mx + c then gives the y-coordinate of the corresponding point on the line. Thus, when dealing with a specific line, only one of the three letters is a true variable.

1.3. BASICS OF POLYNOMIALS 13 Parameters Certain variables are often called parameters; here the idea is that a parameter is a variable which is not intended to be pinned down to a specific value, but is expected to move through a preassigned set of values generating interesting quantities. In other words, a parameter is to be thought of as unspecified constant in a certain range of values. Often, a parameter is described by the oxymoron a variable constant, to remind us of this feature. In the above example of mx + c we may think of m, c as parameters. They can change, but once a line is pinned down, they are fixed. Then only the x stays a variable. As another example, let 1 + 2t be the distance of a particle from a starting position at time t, then t might be thought of as a parameter describing the motion of the particle. It is pinned down as soon as we locate the moving point. The same t can also be simply called a variable if we start asking a question like At what time is the distance equal to 5?. Then we will write an equation 1 + 2t = 5 and by algebraic manipulation, declare that t = 2. Indeterminates An indeterminate usually is a symbol which is not intended to be substituted with values; thus it is like a variable in appearance, but we don t care to assign or change values for it. Thus the statement X 2 a 2 = (X a)(x + a) is a formal statement about expansion and X, a can be thought of as indeterminates. When we use it to factor y 2 9 as (y 3)(y + 3) we are turning X, a into variables and substituting values y, 3 to them! 7 1.3 Basics of Polynomials We now show how polynomials are constructed and handled. This material is very important for future success in Algebra. Monomials Suppose that we are given a variable x, a constant c and a non negative integer n. Then the expression cx n is said to be a monomial. The constant c is said to be its coefficient.(sometimes we call it the coefficient of x n ). 7 In short, the distinction between an indeterminate, variable and parameter, is, like beauty, in the eye of the beholder!

14 CHAPTER 1. REVIEW OF BASIC TOOLS. If the coefficient c is non zero, then the degree of the monomial is defined to be n. If c = 0, then we get the zero monomial and its degree is undefined. It is worth noting that n is allowed to be zero, so 2 is also a monomial with coefficient 2 and degree 0 in any variable (or variables) of your choice! Sometimes, we need monomials with more general exponents, meaning we allow n to be a negative integer as well. The rest of the definition is the same. Sometimes, we may even allow n to be a positive or negative fraction or even a real number. Formally, this is easy, but it takes some effort and great care to give a meaning to the resulting quantity. We will always make it clear if we are using such general exponents. For example, consider the monomial 4x 3. Its variable is x, its coefficient is 4 and its degree is 3. Consider another example 3 2 x5. Its variable is x, its coefficient is 2 3 and its degree is 5. Consider a familiar expression for the area of a circle of radius r, namely πr 2. This is a monomial of degree 2 in r with coefficient π. Thus, even though it is a Greek letter, the symbol π is a well defined fixed real number. 8 This basic definition can be made fancier as needed. Usually, n is an integer, but in higher mathematics, n can be a more general object. As already stated, we can accept 2x 5 3 = ( 5/2)x 3 as an acceptable monomial with general exponents. We declare that its variable is x and it has degree 3 with coefficient 5/2. Notice that we collected all parts (including the minus sign) of the expression except the power of x to build the coefficient! Monomials in several variables. It is permissible to use several variables and write a monomial of the form cx p y q. This is a monomial in x, y with coefficient c. Its degree is said to be p + q. Its exponents are said to be (p, q) respectively (with respect to variables x, y). As before, more general exponents may be allowed, if necessary. 9 8 Note that this famous number has a long history associated with it and people have spent enormous energy in determining its decimal expansion to higher and higher accuracy. There is no hope of ever listing all the infinitely many digits, since they cannot have any repeating pattern because the number is not rational. Indeed, it has been proved to be transcendental, meaning it is not the solution to any polynomial equation with integer coefficients! It is certainly very interesting to know more about this famous number! 9 For the reader who likes formalism, here is a general definition. A monomial in variables x 1,, x r is an expression of the form cx n1 x nr where n 1,, n r are non negative integers and c is any expression which is free of the variables x 1,, x r. The coefficient of the monomial is c, its degree is n 1 + + n r and the exponents are said to be n 1,, n r. If convenient, n 1,, n r may be allowed to be more general.

1.3. BASICS OF POLYNOMIALS 15 Example. Consider the following expression which becomes a monomial after simplification. expression in x,y. 21x 3 y 4 3xy. First, we must simplify the expression thus: 21x 3 y 4 3xy = 21 3 x3 x y4 y = 7x2 y 3. Now it is a monomial in x, y. What are the exponents, degree etc.? Its coefficient is 7, its degree is 2 + 3 = 5 and the exponents are (2, 3). What happens if we change our idea of who the variables are? Consider the same simplified monomial 7x 2 y 3. Let us think of it as a monomial in y. Then we rewrite it as: ( 7x 2 )y 3. Thus as a monomial in y alone, its coefficient is 7x 2 and its degree is 3. What happens if we take the same monomial 7x 2 y 3 and make x as our variable? Then we rewrite it as: ( 7y 3 )x 2. Be sure to note the rearrangement! Thus as a monomial in x alone, its coefficient is 7y 3 and its degree is 2. Here is yet another example. Consider the monomial 2x y = 2x1 y ( 1). Think of this as a monomial in x, y with more general exponents allowed. As a monomial in x, y its degree is 1 1 = 0 and its coefficient is 2. Its exponents are (1, 1) respectively. Binomials, Trinomials, and Polynomials We now use the monomials as building blocks to build other algebraic structures. When we add monomials together, we create binomials, trinomials, and more generally polynomials. It is easy to understand these terms by noting that the prefix mono means one. The prefix bi means two, so it is a sum of two monomials. The prefix tri means three, so it is a sum of three monomials. 10 10 Can you find or make up names for a polynomial with four or five or six terms?

16 CHAPTER 1. REVIEW OF BASIC TOOLS. In general poly means many so we define: Definition: Polynomial A polynomial is defined as a sum of finitely many monomials whose exponents are all non negative. Thus f(x) = x 5 + 2x 3 x + 5 is a polynomial. The four monomials x 5, 2x 3, x, 5 are said to be the terms of the polynomial f(x). Usually, a polynomial is defined to be simplified if all terms with the same exponent are combined into a single term. For example, our polynomial is the simplified form of x 5 + x 3 + x 3 + 3x 4x + 5 and many others. An important convention to note: Before you apply any definitions like the degree and the coefficients etc., you should make sure that you have collected all like terms together and identified the terms which are non zero after this collection. For example, f(x) above can also be written as f(x) = x 5 + 2x 3 + x 2 x + 5 x 2 but we don t count x 2 or x 2 among its terms! A polynomial with no terms has to admitted for algebraic reasons and it is denoted by just 0 and called the zero polynomial. About the notation for a polynomial: When we wish to identify the main variable of a polynomial, we include it in the notation. If several variables are involved, several variables may be mentioned. Thus we may write: p(x) = x 3 + ax + 5, h(x, y) = x 2 + y 2 r 2 and so on. With this notation, all variables other than the ones mentioned in the left hand side are treated as constants for that polynomial. The coefficient of a specific power (or exponent) of the variable in a polynomial is the coefficient of the corresponding monomial, provided we have collected all the monomials having the same exponent. Thus the coefficient of x 3 in f(x) = x 5 + 2x 3 x + 5 is 2. Sometimes, we find it convenient to say that the x 3 term of f(x) is 2x 3. The coefficient of the missing monomial x 4 in f(x) is declared to be 0. The coefficient of x 100 in f(x) is also 0 by the same reasoning. Indeed, we can think of infinitely many monomials which have zero coefficients in a given polynomial. They are not to be counted among the terms of the polynomial. What is the coefficient of x 3 in x 4 + x 3 1 2x 3. Remember that we must collect like terms first to rewrite x 4 + (x 3 2x 3 ) 1 = x 4 x 3 1 and then we see that the coefficient is 1.

1.4. WORKING WITH POLYNOMIALS 17 1.3.1 Rational functions. Just as we create rational numbers from ratios of two integers, we create rational functions from ratios of two polynomials. Definition: Rational function A rational function is a ratio of two polynomials p(x) where q(x) is assumed to be a non zero polynomial. q(x) Examples. 2x + 1 x 5, 1 x 3 + x, x3 + x = x 3 + x. 1 Suppose we have rational functions h 1 (x) = p 1(x) q 1 (x) and h 2(x) = p 2(x) q 2 (x). We define algebraic operations of rational functions, just as in rational numbers. and h 1 (x) ± h 2 (x) = p 1(x)q 2 (x) ± p 2 (x)q 1 (x) q 1 (x)q 2 (x) h 1 (x)h 2 (x) = p 1(x)p 2 (x) q 1 (x)q 2 (x). It is easy to deduce that h 1 (x) = h 2 (x) or h 1 (x) h 2 (x) = 0 if and only if the the numerator of h 1 (x) h 2 (x) is zero, i.e. p 1 (x)q 2 (x) p 2 (x)q 1 (x) = 0. For a rational number, we know that multiplying the numerator and the denominator by the same non zero integer does not change its value. (For example 4 = 8.) 5 10 Similarly, we have: p(x) q(x) = d(x)p(x) d(x)q(x) for any non zero polynomial d(x). (For example x x+1 = x2 x x 2 1.) 1.4 Working with polynomials Given several polynomials, we can perform the usual operations of addition, subtraction and multiplication on them. We can also do division, but if we expect the answer to be a polynomial again, then we have a problem. We will discuss these matters later. 11 7 2 11 The situation is similar to integers. If we wish to divide 7 by 2 then we get a rational number which is not an integer any more. It is fine if you are willing to work with the rational numbers.

18 CHAPTER 1. REVIEW OF BASIC TOOLS. As a simple example, let Then f(x) = 3x 5 + x, g(x) = 2x 5 2x 2, h(x) = 3, and w(x) = 2. f(x)+g(x) = 3x 5 +x+2x 5 2x 2 and after collecting terms f(x)+g(x) = 5x 5 2x 2 +x. Notice that we have also arranged the monomials in decreasing degrees, this is a recommended practice for polynomials. What is w(x)f(x) h(x)g(x)? We see that (2)(3x 5 + x) (3)(2x 5 2x 2 ) = 6x 5 + 2x 6x 5 + 6x 2 = 6x 2 + 2x. Definition: Degree of a polynomial with respect to a variable x is defined to be the highest degree of any monomial present in the simplified form of the polynomial (i.e. any term cx m with c 0 in the simplified form of the polynomial). We shall write deg x (p) for the degree of p with respect to x. The degree of a zero polynomial is defined differently by different people. Some declare it not to have a degree, others take it as 1 and yet others take it as. We shall declare it undefined and hence we always have to be careful to determine if our polynomial reduces to 0. We shall use: Definition: Leading coefficient of a polynomial For a polynomial p(x) with degree n in x, by its leading coefficient, we mean the coefficient of x n in the polynomial. Thus given a polynomial x 3 + 2x 5 x 1 we first rewrite it as 2x 5 + x 3 x 1 and then we can say that its degree is 5, and its leading coefficient is 2. The following are some of the evident facts about polynomials and their degrees. Assume that u = ax n +, v = bx m + are non zero polynomials in x of degrees n,m respectively. We shall need the following important observations. 1. If we say that u has degree n, then we mean that the coefficient a of x n is non zero and that none of the other (monomial) terms has degree as high as n. Similarly, if we say that v has degree m, then we mean that b 0 and no other terms in v are of degree as high as m. You might have also seen the idea of division with remainder which says divide 7 by 2 and you get a quotient of 3 and a remainder of 1, i.e. 7 = (2)(3) + 1. As promised, we shall take this up later.

1.4. WORKING WITH POLYNOMIALS 19 2. If c is a non zero number, then cu = cax n + has the same degree n and its leading coefficient is ca. For example, if c is a non zero number, then the degree of c(2x 5 + x 2) is always 5 and the leading coefficient is (c)(2) = 2c. For c = 0 the degree becomes undefined! 3. Suppose that the degrees n, m of u, v are unequal. Given constants c, d, what is the degree of cu + dv? If c, d are non zero, then the degree of cu + dv is the maximum of n, m. If one of c, d is zero, then we have to look closely. For example, if u = 2x 5 + x 2 and v = x 3 + 3x 1, then the degree of cu + dv is determined thus: cu+dv = c(2x 5 +x 2)+d( x 3 +3x 1) = (2c)x 5 +( d)x 3 +(c+3d)x+( 2c d) Thus, if c 0, then degree will be 5. The leading coefficient is 2c. If c = 0 but d 0, then cu + dv = ( d)x 3 + (3d)x + ( d) and clearly has degree 3. The leading coefficient is d. If c, d are both zero then cu + dv = 0 and the degree becomes undefined. 4. Now suppose that the degrees of u, v are the same, i.e. n = m. Let c, d be constants and consider h = cu + dv. Then the degree of h needs a careful analysis. Since m = n, we see that h = (ca + db)x n + and hence we have: either h = 0 and hence deg x (h) is undefined, or 0 deg x (h) n = m.

20 CHAPTER 1. REVIEW OF BASIC TOOLS. For example, let u = x 3 + 3x 2 + x 1 and v = x 3 3x 2 + 2x 2. Calculate cu + dv with c, d constants. h = cu + dv = c( x 3 + 3x 2 + x 1) + d(x 3 3x 2 + 2x 2) = ( c + d)x 3 + (3c 3d)x 2 + (c + 2d)x + ( c 2d) We determine the degree of the expression thus: If c + d 0, i.e. c d then the degree is 3. If c = d, then the x 2 term also vanishes and we get cu + dv = (c + 2d)x + ( c 2d) = (3d)x + ( 3d). Thus if d = 0 then the degree is undefined and if d 0 then the degree is 1. You are encouraged to figure out the formula for the leading coefficients in each case. You should also experiment with other polynomials. 5. The product rule for degrees. The degree of the product of two non zero polynomials is always the sum of their degrees. This means: Indeed, we can see that deg x (uv) = deg x (u) + deg x (v) = m + n. uv = (ax n + terms of degree less than n )(bx m + terms of degree less than m ) = (ab)x (m+n) + terms of degree less than m + n. The leading coefficient of the product is ab, i.e. the product of their leading coefficients. Thus we have an interesting principle. The degree and the leading coefficient of a product of two non zero polynomials can be calculated by ignoring all but the leading terms in each! By repeated application of this principle, we can see the The power rule for degrees. If u = ax n + is a polynomial of degree n in x, then for any positive integer d, the polynomial u d has degree dn and leading coefficient a d. You should think and convince yourself that it is natural to define u 0 = 1 for all non zero polynomials u. Thus, the same formula can be assumed to hold for d = 0.

1.5. EXAMPLES OF POLYNOMIAL OPERATIONS. 21 1.5 Examples of polynomial operations. Example 1: Consider p(x) = x 3 + x + 1 and q(x) = x 3 + 3x 2 + 5x + 5. What are their degrees? Also, calculate the following expressions: and their degrees. p(x) + q(x), p(x) + 2q(x) and p(x) 2 Determine the leading coefficients for each of the answers. Answers: The degrees of p(x), q(x) are both 3 and their leading coefficients are respectively 1, 1. The remaining answers are: p(x) + q(x) = x 3 (1 1) + x 2 (0 + 3) + x(1 + 5) + (1 + 5) = 3x 2 + 6x + 6 p(x) + 2q(x) = x 3 (1 2) + x 2 (0 + 6) + x(1 + 10) + (1 + 10) = x 3 + 6x 2 + 11x + 11 (x 3 + x + 1)(x 3 + x + 1) = (x 6 + x 4 + x 3 ) + (x 4 + x 2 + x) + (x 3 + x + 1) = x 6 + 2x 4 + 2x 3 + x 2 + 2x + 1 The respective degrees are, 2, 3, 6. The leading coefficients are, respectively, 3, 1, 1. Here is an alternate technique for finding the last answer. It will be useful for future work. First, note that the degree of the answer is 3 + 3 = 6 from the product rule above. Thus we need to calculate coefficients of all powers x i for i = 0 to i = 6. Each x i term comes from multiplying an x j term and an x i j term from p(x). Naturally, both j and i j need to be between 0 and 3. Thus the only way to get x 0 in the answer is to take j = 0 and i j = 0 0 = 0. Thus the x 0 term of the answer is the square of the x 0 term of p(x), hence is 1 2 = 1. Similarly, the x 6 term can only come from x 3 term multiplied by the x 3 term and is x 6. Now for the x 5 term, the choices are j = 3, i j = 2 or j = 2, i j = 3 and in p(x) the coefficient of x 2 is 0, so both these terms are 0. Hence the x 5 term is missing in the answer.

22 CHAPTER 1. REVIEW OF BASIC TOOLS. Now for x 4 terms, we have three choices for (j, i j), namely (3, 1), (2, 2), (1, 3). We get the corresponding coefficients (1)(1) + (0)(0) + (1)(1) = 2. The reader should verify the rest. Example 2: Use the above technique to answer the following: Determine a, b, c, d, e. Assume (x 20 + 2x 19 x 17 + + 3x + 5) (3x 20 + 6x 19 4x 18 + + 5x + 6) = ax 40 + bx 39 + cx 38 + + dx + e Note that the middle terms are not even given and not needed for the required answers. The only term contributing to x 40 is (x 20 ) (3x 20 ), so a = 3. The only terms producing x 39 are (x 20 ) (6x 19 ) and (2x 19 ) (3x 20 ) and hence, the coefficient for x 39 is (1)(6) + (2)(3) = 12, so b = 12. Similarly, check that c = (1)( 4) + (2)(6) + (0)(3) = 8. Calculate d = (3)(6) + (5)(5) = 43 and e = (5)(6) = 30. Example 3: The technique of polynomial multiplication can often be used to help with integer calculations. Remember that a number like 5265 is really a polynomial 5d 3 + 2d 2 + 6d + 5 where d = 10. Now, we can use our technique of polynomial operations to add or multiply numbers. The only thing to watch out for is that since d is a number and not really a variable, there are carries to worry about. Here we derive the well known formula for squaring a number ending in 5. Before describing the formula, let us give an example. Say, you want to square the number 25. Split the number as 2 5. From the left part 2 construct the number (2)(3) = 6. This is obtained by multiplying the left part with itself increased by 1. Simply write 25 next to the current calculation, so the answer is 625. The square of 15 by the same technique shall be obtained thus: Split it as 1 5. (1)(1 + 1) = 2. So the answer is 225. The general rule is this: Let a number n be written as p 5 where p is the part of the number after the units digit 5.

1.5. EXAMPLES OF POLYNOMIAL OPERATIONS. 23 Then n 2 = (p)(p + 1)25, i.e. (p)(p + 1) becomes the part of the number from the 100s digit onwards. To illustrate, consider 45 2. Here p = 4, so (p)(p + 1) = (4)(5) = 20 and so the answer is 2025. Also 105 2 = (10)(11)25 = 11025. Similarly 5265 2 = (526)(527)25, i.e. 27720225. What is the proof? Since n splits as p 5, our n = pd + 5. Using that d = 10 we make the following calculations. n 2 = (pd + 5)(pd + 5) = p 2 d 2 + (2)(5)(pd) + 25 = p 2 (100) + p(100) + 25 = (p 2 + p)100 + 25. Thus the part after the 100s digits is p 2 + p = p(p + 1). You can use such techniques for developing fast calculation methods. Example 4: Let us begin by f(x) = x n for n = 1, 2, and let us try to calculate the expansions of f(x + t). For n = 1 we simply get f(x + t) = x + t. For n = 2 we get f(x + t) = (x + t) 2 = x 2 + 2xt + t 2. For n = 3 we get f(x + t) = (x + t) 3. Let us calculate this as follows: (x + t)(x + t) 2 = (x + t)(x 2 + 2xt + t 2 ) = x(x 2 + 2xt + t 2 ) +t(x 2 + 2xt + t 2 ) = x 3 + 2x 2 t + xt 2 + x 2 t + 2xt 2 + t 3 = x 3 + 3x 2 t + 3xt 2 + t 3 For n = 4 we invite the reader to do a similar calculation and deduce 12 (x + t) 4 = (x + t)(x 3 + 3x 2 t + 3xt 2 + t 3 ) = x 4 + (3x 3 t + x 3 t) + (3x 2 t 2 + 3x 2 t 2 ) + (xt 3 + 3xt 3 ) + t 4 = x 4 + 4x 3 t + 6x 2 t 2 + 4xt 3 + t 4. 12 An alert reader may see similarities with the process of multiplying two integers by using one digit of the multiplier at a time and then adding up the resulting rows of integers. Indeed, it is the result of thinking of the numbers as polynomials in 10, but we have to worry about carries. Here is 4 3 1 1 2 a sample multiplication for your understanding: 8 6 2 4 3 1 5 1 7 2